What is the integral of (2x)/(1+x^4) ?

The integral of (2x)/(1+x^4) is arctan(x^2)+C

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integral of (2x)/(1+x^4)

\int \frac{2 x}{1 + x ^{4}} d x

arctan (x^{2}) + C

Solution steps

\int \frac{2 x}{1 + x ^{4}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{x}{1 + x ^{4}} d x

Apply u-substitution

: \int \frac{1}{2 ( 1 + u ^{2} )} d u

= 2 \cdot \int \frac{1}{2 ( 1 + u ^{2} )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= 2 \cdot \frac{1}{2} arctan (u)

Substitute back

u = x^{2}

= 2 \cdot \frac{1}{2} arctan (x^{2})

Simplify

2 \cdot \frac{1}{2} arctan (x^{2}) : arctan (x^{2})

= arctan (x^{2})

Add a constant to the solution

= arctan (x^{2}) + C

y = (x + 3)^{3}

x^{2} y^{^{''}} + x y^{^{'}} - y = \frac{ln ( x )}{x ^{2}}

\frac{x + 1}{x ^{2} + 1}

y^{^{'}} = 4 y + 1

What is the integral of (2x)/(1+x^4) ?
The integral of (2x)/(1+x^4) is arctan(x^2)+C